Triangle rendering using direct evaluation

ABSTRACT

This disclosure describes an apparatus, such as a wireless communication device, that applies a direct evaluation technique to render triangles for the 3D graphical environment. The apparatus includes a rendering engine that defines a rectangular area of pixels, referred to as a bounding box, that bounds the area to be rendered. The rendering engine evaluates coordinates associated with the pixels of the rectangular area to selectively render those pixels that fall within the triangular area. The direct evaluation triangle rendering algorithm may require fewer complex operations than the more computationally intensive interpolation process employed by other systems. As a result, the apparatus may present a 3D graphical environment while preserving as much as possible the available power.

This application claims priority from U.S. Provisional Application Ser. No. 60/455,923, filed Mar. 18, 2003, the entire content of which is incorporated herein by reference.

TECHNICAL FIELD

The invention relates to mobile computing devices and, in particular, to techniques for rendering graphics for display by the mobile computing devices.

BACKGROUND

Mobile computing devices, such as personal digital assistants (PDAs), wireless communication devices, global positioning devices, and the like, are increasingly requiring advanced two-dimensional (2D) and three-dimensional (3D) graphics applications. For example, the mobile computing devices increasingly offer games, character animations, graphical menu systems, and other applications that require advanced graphics rendering. This advanced graphics rendering, especially with 3D graphics, involves a substantial amount of data processing. Furthermore, these processing requirements only increase as the mobile computing devices adopt display panels that offer increased resolution and color depth.

One fundamental operation of a 3D graphics application is triangular rendering. In particular, a 3D graphical object is typically defined by a set of triangles in a 3D space. Each triangle is defined by three vertices within the 3D space and associated attributes of each vertex, including color attributes, texture coordinates, or both. Each triangle is graphically rendered using an interpolation process that fills the interior of the triangle based on the location and attribute information defined at the three vertices. More specifically, each starting and ending coordinate is computed for each line of pixels within the triangle using the interpolation process. The pixels along each line are sequentially filled to render the entire triangle.

One challenge for mobile computing devices, however, is the typical limitation of available power. Unlike a desktop computing environment, mobile computing devices typically have limited battery power. As a result, the computationally intensive interpolation process required for triangular rendering may place a significant strain on the available power of the mobile computing device.

SUMMARY

The disclosure is directed to techniques for rendering a triangular area of pixels for presentation on a display device. The techniques may be particularly useful in a mobile computing environment presenting limited power resources.

In one embodiment, a method comprises computing data that defines a rectangular area of pixels that bounds a triangular area of the pixels, and evaluating coordinates of the pixels of the rectangular area to determine which pixels fall within the triangle area., The method further comprises updating pixel data for the pixels that fall within the triangle area to render the triangular area.

In another embodiment, an apparatus comprises a rendering engine that defines a rectangular area of pixels that bounds a triangular area of the pixels. The rendering engine evaluates coordinates associated with the pixels of the rectangular area to selectively render the pixels that fall within the triangular area.

In another embodiment, a mobile communication device comprises a display, a processor, and a rendering engine. The processor generates video output data for presentation by the display as a graphical environment. The rendering engine applies a direct evaluation algorithm to render a triangle for the graphical environment, wherein the direct evaluation algorithm applies linear equations to render the triangle without interpolating between edges of the triangle.

The details of one or more embodiments of the invention are set forth in the accompanying drawings and the description below. Other features, objects, and advantages of the invention will be apparent from the description and drawings, and from the claims.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram illustrating an example embodiment of a mobile computing device.

FIG. 2 is a graph that illustrates application of direct evaluation triangle rendering techniques;

FIG. 3 is a flowchart illustrating example operation of a rendering engine in application of the direct evaluation techniques.

FIG. 4 is a block diagram illustrating an example embodiment of the rendering engine of FIG. 3.

DETAILED DESCRIPTION

FIG. 1 is a block diagram illustrating an example mobile computing device 2 that presents a three-dimensional (3D) graphical environment on display 4. Mobile computing device 2 may be, for example, a personal digital assistant (PDA), a wireless communication device such as a mobile phone or satellite phone, a global positioning device, a portable digital television, a digital video camera, a wireless video device, a device integrating any of the foregoing devices, and the like. Examples of the 3D graphical environment presented by mobile computing device 2 include games, character animations, graphical menu systems, and other applications that require advanced graphics rendering.

Processor 6 provides primary control over the components of mobile computing device 2. For example, processor 6 captures input from a user via input device 8, such as a keypad, and directs digital signal processor (DSP) 10 to generate video output data for presenting the 3D graphical environment on display 4. Display 4 may comprise any output device, e.g., a liquid crystal display (LCD) for a camcorder or a mobile phone screen. In particular, DSP 10 issues commands to rendering engine 12 to direct the rendering engine to render graphical objects, e.g., triangles, within a 3D space. Based on the commands, rendering engine 12 processes video data stored within video memory 14 and cache 15 for driving display 4.

Processor 6 may take the form of an embedded microprocessor, specialized hardware, software, e.g., a control software module, or combinations thereof. Moreover, DSP 10, processor 6, rendering engine 12, as well as other components of mobile computing, device 2, may be implemented in one or more application-specific integrated circuits (ASICs), as multiple discrete components, or combinations thereof.

Memories 16 and 18 store instructions and data for use by processor 6 and DSP 10, respectively. Although illustrated separately, memories 16 and 18 may comprise one or more memory devices. Moreover, memories 16 and 18 may comprise read-only memory (ROM), synchronous dynamic random access memory (SDRAM), non-volatile static random access memory (SRAM), Flash memory, electrically erasable programmable read-only memory (EEPROM), and the like.

As described herein, DSP 10 and a rendering engine 12 interact to apply a direct evaluation technique to render triangles for the 3D graphical environment. More specifically, DSP 10 issues commands to rendering engine 12 to direct the rendering engine to render triangles within a 3D space. In response, rendering engine 12 applies a direct evaluation triangle rendering algorithm that requires fewer division operations than the more computationally intensive interpolation process employed by conventional systems. As a result, mobile computing device 2 may present a 3D graphical environment while preserving as much as possible the power available to the mobile computing device.

To invoke rendering engine 12, DSP 10 issues a command that specifies the three vertices of the triangle to be rendered. For example, as illustrated in the example triangle depicted by FIG. 2, DSP 10 issues a command that defines triangle 20 with a first vertex (X₀,Y₀), a second vertex (X₁,Y₁), and a third vertex (X₂,Y₂), where each of the vertices represents a coordinate within a 2D slice of a 3D environment. Consequently, the vertices define a triangle having edges 24A, 24B and 24C.

In response to the command, rendering engine 12 computes a “bounding box” 22, which refers to a rectangular area defined to bound a minimum and a maximum of the x,y coordinates of the triangle. Bounding box 22, for example, is defined by an upper-left coordinate (X_(min), Y_(min)) and a lower-right coordinate (X_(max), Y_(max)). As illustrated in the, example of FIG. 2, bounding box 22 may be defined to cover a minimum area within the 2D slice to encompass triangle 20. In other embodiments, minimum dimensions for bounding box 22 may be computed as function of a block size of video blocks stored by cache 15. In particular, DSP 10 and rendering engine 12 may process output video data of video memory 14 in a “tiled” fashion, i.e., in block fashion along rows and columns. Moreover, the block size may be set based on the capacity of cache 15. For example, the block size may equal the capacity of the cache. In this manner, the dimensions of bounding box 22 may be defined to be equal to or less than the dimensions of a video block in the event a triangle to be rendered lies completely within a single video block. For triangles spanning more than one video block, a bounding box may be used that has dimensions calculated to encompass the overlap between the bounding box of the triangle and the video block, as illustrated by bounding box 22A. This may advantageously reduce the number of memory accesses required for retrieving and rendering pixel data associated with the triangle, which translates into power savings.

After calculating the bounding box, e.g., bounding box 22 of FIG. 2, rendering engine 12 processes each pixel line encompassed by the bounding box and selectively enables those pixels that fall within triangle 20. Specifically, rendering engine 12 scans the pixels lines, e.g., in a rightward and downward fashion, and applies linear equations to determine whether each pixel falls within triangle 20. For those pixels within the edges 24A, 24B and 24C, rendering engine 12 updates the current attributes for the pixels, e.g., based on z-value, color information, texture information, or the like, associated with triangle 20.

FIG. 3 is a flowchart illustrating example operation of rendering engine 12 in application of a direct evaluation algorithm for rendering a triangle. Initially, rendering engine 12 receives a command from DSP 10 that specifies three vertices of a triangle to be rendered, i.e., a first vertex (X₀,Y₀), a second vertex (X₁,Y₁), and a third vertex (X₂,Y₂) (step 30 of FIG. 3).

In response to the command, rendering engine 12 computes data, e.g., an upper left coordinate and a height and width, that defines the bounding box that encompasses the triangle:defined by the vertices (step 32).

In addition, rendering engine 12 computes a coefficient matrix M⁻¹ for computing the linear coefficients of the linear equations that describe the attributes of each pixel within the triangle, and a coefficient matrix M_(C) for computing the linear coefficients of the linear equations that describe the three edges of the triangle (step 34). Specifically, an attribute value v for a pixel located inside the triangle can be represented as a linear function of the x,y location in the screen space as follows: v=Ax+By+C.  (1)

Consequently, the linear coefficients, A, B, and C for eq. (1) can be obtained from the three equations associated with the three vertices: v ₀ =Ax ₀ +By ₀ +C v ₁ =Ax ₁ +By ₂ +C v ₂ =Ax ₂ +By ₂ +C  (2) where v_(i)(i=0, 1, 2), is the attribute value defined at vertex i, and (x_(i),y_(i))(i=0, 1, 2) are the screen coordinates of vertex i. Using matrix notations, the vertices can be represented as:

$\begin{matrix} {\begin{bmatrix} v_{0} \\ v_{1} \\ v_{2} \end{bmatrix} = {\begin{bmatrix} x_{0} & y_{0} & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{bmatrix} = {\begin{bmatrix} A \\ B \\ C \end{bmatrix}.}}} & (3) \end{matrix}$

Thus, the coefficients for attribute interpolation can be expressed as:

$\begin{matrix} {\begin{bmatrix} A \\ B \\ C \end{bmatrix} = {{M^{- 1}\begin{bmatrix} v_{0} \\ v_{1} \\ v_{2} \end{bmatrix}}.}} & (4) \end{matrix}$ where

$\begin{matrix} {M = \begin{bmatrix} x_{0} & y_{0} & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{bmatrix}} & (5) \end{matrix}$

The coefficient matrix M⁻¹ for computing the coefficients for attribute interpolation can be computed using Laplace expansion as follows

$\begin{matrix} {M^{- 1} = {\frac{1}{\det(M)}M_{C}^{T}}} & (6) \end{matrix}$ where det(M)=|M|=x ₁ y ₂ +x ₂ y ₀ +x ₀ y ₁ −x ₂ y ₁ −x ₀ y ₂ −x ₁ y ₀  (7) M_(C), the companion matrix of M, can be calculated as:

$\begin{matrix} {M_{c} = {\begin{bmatrix} {y_{1} - y_{2}} & {x_{2} - x_{1}} & {{x_{1}y_{2}} - {x_{2}y_{1}}} \\ {y_{2} - y_{0}} & {x_{0} - x_{2}} & {{x_{2}y_{0}} - {x_{0}y_{2}}} \\ {y_{0} - y_{1}} & {x_{1} - x_{0}} & {{x_{0}y_{1}} - {x_{1}y_{0}}} \end{bmatrix}.}} & (8) \end{matrix}$

Upon calculating the coefficient matrices M⁻¹ and M_(C), rendering engine 12 computes coefficients A, B, C, for all attributes associated with the pixels (step 36). In particular, the coefficient matrix M⁻¹ can be used for computing the coefficients of different attributes, although the values at the vertices, (v₀,v₁,v₂), are different for different attributes.

Next, rendering engine 12 initializes all attribute values at a starting pixel of the bounding box, e.g., (X_(min), Y_(min)) of FIG. 2, (step 37). In addition, rendering engine 12 initializes current pixel index values (X_(C), Y_(C)) to a first pixel of a first scan line of the bounding box, i.e., X_(MIN) and Y_(MIN) (step 38).

During the direct evaluation rasterization process, rendering engine 12 tests the current pixel (X_(C), Y_(C)) to determine whether the pixel falls within the triangle (step 39). Specifically, rendering engine 12 applies the previously calculated matrix M_(C) based on a set of edge equations delineating the triangle to determine whether the location of the current pixel falls within the triangle. These equations are uniquely defined by the coordinates of the three vertices. In particular, for a given starting vertex (x_(i), y_(i)), and ending point (x_(j), y_(j)), on a 2D plane, for example, the edge equation is,

$\begin{matrix} {\frac{y - y_{i}}{x - x_{i}} = \frac{y_{j} - y_{i}}{x_{j} - x_{i}}} & (9) \end{matrix}$ or, (y _(i) −y _(j))x+(x _(j) −x _(i))y+x _(i) y _(j) −x _(j) y=0.  (10) For the three edges with starting vertex index, i=1, 2, 0, and ending vertex index, j=2, 0, 1, three edge equations in matrix format can be expressed as:

$\begin{matrix} {\begin{bmatrix} e_{1} \\ e_{2} \\ e_{3} \end{bmatrix} = {{M_{C}\begin{bmatrix} X_{C} \\ Y_{C} \\ 1 \end{bmatrix}} = {\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.}}} & (11) \end{matrix}$

Thus, rendering engine 12 applies the matrix M_(C) to the current pixel (X_(C), Y_(C)), and determines that the pixel is inside the triangle when all of e₁, e₂, e₃ are less than zero.

If the pixel is not within the triangle (no branch of 39), rendering engine 12 proceeds to the next pixel along the scan line of the bounding box, i.e., increments X_(C) (step 46) and, assuming the end of the scan line has not been reached (step 48), tests that pixel in accordance with equation (11) to determine whether the pixel is within the triangle (step 39).

If application of equation (11) indicates that the current pixel is within the triangle, rendering engine 12 determines whether the current pixel is visible (step 40). Specifically, rendering engine 12 determines the position of the current pixel along the z-axis to determine whether the pixels is visible, or whether pixels from other slices of the 3D graphic environment prevent the pixel from being visible to the user, i.e., are on top of the current pixel. In one embodiment, rendering engine 12 compares a z-value, z_(c), of the current pixel with the corresponding z-value, z_(b), of a z-buffer to determine if the pixel is visible, i.e., if z_(c)<z_(b).

If the pixel is not visible (no branch of step 40), rendering engine 12 proceeds to the next pixel (step 46). If, however, the pixel is visible, rendering engine 12 computes attribute values, e.g., texture and color values, for the pixel based on equation (1) (step 42). Upon computing the attribute values, rendering engine 12 updates the value of the current pixel within video memory 14 based on the computed attributes (step 44).

After updating the value of the current pixel within video memory 14, rendering engine 12 proceeds to the next pixel (step 46). This process is repeated until the end of the current scan line is reached, i.e., X_(C)>X_(MAX) or the current pixel exits the triangle (step 48), causing rendering engine 12 to proceed to the beginning of the next scan line, i.e., set X_(C)=X_(MIN) and incrementing Y_(C) (step 50). Rendering engine 12 traverses each scan line of the bounding box testing each pixel in this manner, and terminates the rendering process when the last scan line has been completed, i.e., Y_(C)>Y_(MAX) (step 52).

FIG. 4 is a block diagram illustrating an example embodiment of rendering engine 12. In the illustrated embodiment, rendering engine 12 includes a vertex buffer 60 for buffering vertices for triangles to be rendered in accordance with the direct evaluation techniques. In particular, rendering engine 12 receives commands from DSP 10 that specify triangles in the form of three vertices. Each vertex is defined by at least an x and y coordinate, and may include one or more attribute values k, such as z-values, color values, texture values, and the like.

Rendering engine 12 queues the vertex information in vertex buffer 60 for processing by processing engine 61, which includes bounding box generator 62, edge coefficient generator 64, and attribute coefficient generator 66. Bounding box generator 62 computes and outputs data values that define the dimensions of the bounding box associated with the triangle to be rendered. For example, bounding box generator 62 may output an upper-left coordinate (X_(min), Y_(min)) and a lower-right coordinate (X_(max), Y_(max)) for the bounding box.

Edge coefficient generator 64 processes the vertex coordinate information to generate the linear edge coefficients matrix M_(c), as described above. Similarly, attribute coefficient generator 66 processes the vertex coordinate information to generate the linear attribute coefficients A, B, C as described above.

Rendering engine 12 further includes rasterizer 74 that processes the parameters produced by processing engine 61, and applies the direct evaluation techniques to produce pixel data to drive video memory 14 and display 4. Output buffer 63 includes bounding box buffer 68, edge coefficient buffer 70, and attribute coefficient buffer 72 to queue parameters defining the bounding box, linear edge coefficients, and linear attribute coefficients associated with each triangle to be rendered by rasterizer 74.

Various embodiments of the invention have been described. These and other embodiments are within the scope of the following claims. 

1. An apparatus comprising: a rendering engine that defines a rectangular area of pixels that bounds an entire triangular area of the pixel that defines a triangle to be rendered, wherein the rectangular area of pixels includes one or more lines of pixels; and the rendering engine further selects each of the one or more lines of pixels within the rectangular area of pixels, sequentially evaluates coordinates associated with the pixels of each line of pixels starting at one end of the rectangular area to determine whether the one or more pixels fall within the triangular area, wherein the one end of the rectangular area is common for the sequential evaluation of each line of pixels, ceases evaluation of the coordinates associated with the pixels of each line of pixels upon determining that at least one pixel of the line falls within the triangular area and a current pixel no longer falls within the triangular area, stores information indicating which of the pixels fall within the triangular area, selectively renders the pixels that fall within the triangular area by computing updated pixel data for those pixels in accordance with a set of linear equations that describe one or more attributes associated with the triangular area, and computes an inverse coefficient matrix M¹ for computing linear coefficients A, B, C of the set of linear equations, and applies the coefficient A, B, C to each pixel that falls within the triangular area to compute an attribute value for the respective pixel, wherein the rendering engine applies the coefficient matrix M¹ to compute the linear coefficients A, B, C, for an attribute associated with vertices v₀(x₀,y₀), v₁(x₁,y₁), and v₂(x₂,y₂) of the triangle as: ${\begin{bmatrix} A \\ B \\ C \end{bmatrix} = {M^{- 1}\begin{bmatrix} v_{0} \\ v_{1} \\ v_{2} \end{bmatrix}}},$ where a coefficient matrix M equals: ${M = \begin{bmatrix} x_{0} & y_{0} & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{bmatrix}},$ where the inverse coefficient matrix M¹ equals: ${M^{- 1} = {\frac{1}{\det(M)}M_{C}^{T}}},$ where det(M) equals: det(M)=|M|=x ₁ y ₂ +x ₂ y ₀ +x ₀ y ₁ −x ₂ y ₁ −x ₀ y ₂ −x ₁ y ₀, where M^(T) _(C) is a transpose of matrix M_(C); where matrix M_(C) equals: ${M_{c} = \begin{bmatrix} {y_{1} - y_{2}} & {x_{2} - x_{1}} & {{x_{1}y_{2}} - {x_{2}y_{1}}} \\ {y_{2} - y_{0}} & {x_{0} - x_{2}} & {{x_{2}y_{0}} - {x_{0}y_{2}}} \\ {y_{0} - y_{1}} & {x_{1} - x_{0}} & {{x_{0}y_{1}} - {x_{1}y_{0}}} \end{bmatrix}}\;,\;{and}$ an attribute value for each pixel (X_(c), Y_(c)) is computed as v=AX _(c) +BY _(c) +C.
 2. The apparatus of claim 1, wherein the rendering engine evaluates the coordinates of the pixels in accordance with a set of linear equations that describe edges of the triangular area.
 3. The apparatus of claim 2, wherein the rendering engine computes the coefficient matrix M_(C) for computing linear coefficients for the set of linear equations, and applies the coefficient matrix M_(C) to one or more pixels within the rectangular area to determine whether each of the one or more pixels falls within the triangular area.
 4. The apparatus of claim 3, wherein the rendering engine applies the coefficient matrix M_(C) to a current one of the one or more pixels (X_(C), Y_(C)) within the rectangular area, wherein the rendering engine determines that the current one of the one or more pixels is within the triangular area when ${M_{C}\begin{bmatrix} X_{C} \\ Y_{C} \\ 1 \end{bmatrix}}{< {\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.}}$
 5. The apparatus of claim 1, wherein the attribute values comprise at least one of color values and texture values.
 6. The apparatus of claim 1, further comprising a z-buffer storing a set of z-values associated with the pixels, and wherein the rendering engine compares a z-value, z_(c), of the current pixel with a corresponding z-value, z_(b), of a z-buffer to determine whether each pixel within the rectangular area is visible and selectively renders each pixel of the rectangular area that is visible and that falls within the triangle area.
 7. The apparatus of claim 1, further comprising a control unit that issues a command to the rendering engine that specifies vertices of the triangular area.
 8. The apparatus of claim 1, wherein the rendering engine comprises: a vertex buffer for buffering the vertices of the triangular area to be rendered; a bounding box generator that processes the vertices to compute bounding data that define the dimensions of the rectangular area; and a rasterizer that processes the bounding data and evaluates coordinates associated with one or more of the pixel values of the rectangular area to selectively render pixels that fall within the triangular area.
 9. The apparatus of claim 8, further comprising: an edge coefficient generator that receives the vertices buffered by the vertex buffer and processes the vertices to compute linear coefficients for a set of linear equations that describe edges of the triangular area, and an attribute coefficient generator that processes the vertices to compute linear coefficients for a set of linear equations that describe one or more attributes associated with the triangular area, wherein the rasterizer processes the bounding data and the coefficients in accordance with the sets of linear equations to render the pixels that fall within the triangular area.
 10. The apparatus of claim 1, wherein the apparatus comprises a wireless communication device.
 11. The apparatus of claim 1, wherein the apparatus comprises an integrated circuit.
 12. The apparatus of claim 1, further comprising a cache memory to store at least a portion of the pixels within the rectangular area, wherein the cache memory has a block size, and the rendering engine defines the rectangular area as a function of the block size of the cache memory.
 13. The apparatus of claim 1, wherein the rendering engine sequentially evaluates coordinates associated with the pixels of each line of pixels in a rightward and downward fashion.
 14. The apparatus of claim 1, wherein the apparatus comprises a mobile phone.
 15. An apparatus comprising: means for rendering that defines a rectangular area of pixels that bounds an entire triangular area of the pixels that defines a triangle to be rendered, wherein the rectangular area of pixels includes one or more lines of pixels; the means for rendering further selects each of the one or more lines of pixels within the rectangular area of pixels, sequentially evaluates coordinates associated with the pixels of each line of pixels starting at one end of the rectangular area to determine whether the one or more pixels fall within the triangular area, wherein the one end of the rectangular area is common for the sequential evaluation of each line of pixels, ceases evaluation of the coordinates associated with the pixels of each line of pixels upon determining that at least one pixel of the line falls within the triangular area and a current pixel no longer falls within the triangular area, selectively renders the pixels that fall within the triangular area by computing updated pixel data for those pixels in accordance with a set of linear equations that describe one or more attributes associated with the triangular area, and computes an inverse coefficient matrix M¹ for computing linear coefficients A, B, C of the set of linear equations, and applies the coefficients A, B, C to each pixel that falls within the triangular area to compute an attribute value for the respective pixel, wherein the means for rendering applies the coefficient matrix M¹ to compute the linear coefficients A, B, C, for an attribute associated with vertices v₀(x₀,y₀), v₁(x₁,y₁), and v₂(x₂,y₂) of the triangle as: ${\begin{bmatrix} A \\ B \\ C \end{bmatrix} = {M^{- 1}\begin{bmatrix} v_{0} \\ v_{1} \\ v_{2} \end{bmatrix}}},$ where a coefficient matrix M equals: ${M = \begin{bmatrix} x_{0} & y_{0} & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{bmatrix}},$ where the inverse coefficient matrix M¹ equals: ${M^{- 1} = {\frac{1}{\det(M)}M_{C}^{T}}},$ where det(M) equals: det(M)=|M|=x ₁ y ₂ +x ₂ y ₀ +x ₀ y ₁ −x ₂ y ₁ −x ₀ y ₂ −x ₁ y ₀, where M^(T) _(C) is a transpose of matrix M_(C), where matrix M_(C) equals: ${M_{c} = \begin{bmatrix} {y_{1} - y_{2}} & {x_{2} - x_{1}} & {{x_{1}y_{2}} - {x_{2}y_{1}}} \\ {y_{2} - y_{0}} & {x_{0} - x_{2}} & {{x_{2}y_{0}} - {x_{0}y_{2}}} \\ {y_{0} - y_{1}} & {x_{1} - x_{0}} & {{x_{0}y_{1}} - {x_{1}y_{0}}} \end{bmatrix}}\;,\mspace{11mu}{and}$ an attribute value for each pixel (X_(c), Y_(c)) is computed as v=AX _(c) +BY _(c) +C; and means for storing information indicating which of the pixels tail within the triangular area.
 16. The apparatus of claim 15, wherein the means for rendering evaluates the coordinates of the pixels in accordance with a set of linear equations that describe edges of the triangular area.
 17. The apparatus of claim 16, wherein the means for rendering computes the coefficient matrix M_(C) computing linear coefficients for the set of linear equations, and applies the coefficient matrix M_(C) to one or more pixels within the rectangular area to determine whether each of the one or more pixels falls within the triangular area.
 18. The apparatus of claim 17, wherein the means for rendering applies the coefficient matrix M_(C) to a current one of the one or more pixels (X_(C), Y_(C)) within the rectangular area, wherein the means for rendering determines that the current one of the one or more pixels is within the triangular area when ${M_{C}\begin{bmatrix} X_{C} \\ Y_{C} \\ 1 \end{bmatrix}}{< {\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.}}$
 19. The apparatus of claim 15, wherein the attribute values comprise at least one of color values and texture values.
 20. The apparatus of claim 15, further comprising a means for storing a set of z-values associated with the pixels, and wherein the means for rendering compares a z-value, Z_(c), of the current pixel with a corresponding z-value, z_(b), of a z-buffer to determine whether each pixel within the rectangular area is visible and selectively renders each pixel of the rectangular area that is visible and that falls within the triangular area.
 21. The apparatus of claim 15, further comprising a means for controlling that issues a command to the means for rendering that specifies vertices of the triangular area.
 22. The apparatus of claim 15, wherein the means for rendering comprises: means for buffering the vertices of the triangular area to be rendered; means for processing the vertices to compute bounding data that define the dimensions of the rectangular area; and means for processing the bounding data and evaluates coordinates associated with one or more of the pixel values of the rectangular area to selectively render pixels that fall within the triangular area.
 23. The apparatus of claim 22, further comprising: means for receiving the vertices buffered by the means for buffering and means for processing the vertices to compute linear coefficients for a set of linear equations that describe edges of the triangular area, and means for processing the vertices to compute linear coefficients for a set of linear equations that describe one or more attributes associated with the triangular area, wherein the means for processing the bounding data processes the bounding data and the coefficients in accordance with the sets of linear equations to render the pixels that fall within the triangular area.
 24. The apparatus of claim 15, wherein the apparatus comprises a wireless communication device.
 25. The apparatus of claim 15, wherein the apparatus comprises an integrated circuit.
 26. The apparatus of claim 15, further comprising a cache memory to store at least a portion of the pixels within the rectangular area, wherein the cache memory has a block size, and the means for rendering defines the rectangular area as a function of the block size of the cache memory.
 27. The apparatus of claim 15, wherein the means for rendering sequentially evaluates coordinates associated with the pixels of each line of pixels in a rightward and downward fashion.
 28. A computer-readable storage medium comprising one or more memory devices that store instructions that cause one or more processors to: define a rectangular area of pixels that bounds an entire triangular area of the pixels that defines a triangle to be rendered, wherein the rectangular area of pixels includes one or more lines of pixels; select each of the one or more lines of pixels within the rectangular area of pixels; sequentially evaluate coordinates associated with the pixels of each line of pixels starting at one end of the rectangular area to determine whether the one or more pixels fall within the triangular area, wherein the one end of the rectangular area is common for the sequential evaluation of each line of pixels; cease evaluation of the coordinates associated with the pixels of each line of pixels upon determining that at least one pixel of the line falls within the triangular area and a current pixel no longer falls within the triangular area; selectively render the pixels that fall within the triangular area by computing updated pixel data for those pixels in accordance with a set of linear equations that describe one or more attributes associated with the triangular area; compute an inverse coefficient matrix M¹ for computing linear coefficients A, B, C of the set of linear equations, and to apply the coefficients A, B, C to each pixel that falls within the triangular area to compute an attribute value for the respective pixel; apply the coefficient matrix M¹ to compute the linear coefficients A, B, C, for an attribute associated with vertices v₀(x₀,y₀), v₁(x₁,y₁), and v₂(x₂,y₂) of the triangle as: ${\begin{bmatrix} A \\ B \\ C \end{bmatrix} = {M^{- 1}\begin{bmatrix} v_{0} \\ v_{1} \\ v_{2} \end{bmatrix}}},$ where a coefficient matrix M equals: ${M = \begin{bmatrix} x_{0} & y_{0} & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{bmatrix}},$ where the inverse coefficient matrix M¹ equals: ${M^{- 1} = {\frac{1}{\det(M)}M_{C}^{T}}},$ where det(M) equals: det(M)=|M|=x ₁ y ₂ +x ₂ y ₀ +x ₀ y ₁ −x ₂ y ₁ −x ₀ y ₂ −x ₁ y ₀, where M^(T) _(C) is a transpose of matrix M_(C), where matrix M_(C) equals: ${M_{c} = \begin{bmatrix} {y_{1} - y_{2}} & {x_{2} - x_{1}} & {{x_{1}y_{2}} - {x_{2}y_{1}}} \\ {y_{2} - y_{0}} & {x_{0} - x_{2}} & {{x_{2}y_{0}} - {x_{0}y_{2}}} \\ {y_{0} - y_{1}} & {x_{1} - x_{0}} & {{x_{0}y_{1}} - {x_{1}y_{0}}} \end{bmatrix}}\;,\mspace{11mu}{and}$ an attribute value for each pixel (X_(c), Y_(c)) is computed as v=AX _(c) +BY _(c) +C; and store information indicating which of the pixels fall within the triangular area.
 29. The computer readable storage medium of claim 28, wherein the coordinates of the pixels are evaluated in accordance with a set of linear equations that describe edges of the triangular area.
 30. The computer readable storage medium of claim 29, further comprising instructions that cause one or more processors to compute the coefficient matrix M_(C) for computing linear coefficients for the set of linear equations, and to apply the coefficient matrix M_(C) to one or more pixels within the rectangular area to determine whether each of the one or more pixels falls within the triangular area.
 31. The computer readable storage medium of claim 30, further comprising instructions that cause one or more processors to apply the coefficient matrix M_(C) to a current one of the one or more pixels (X_(C), Y_(C)) within the rectangular area, wherein the instructions cause the one or more processors to determine that the current one of the one or more pixels is within the triangular area when ${M_{C}\begin{bmatrix} X_{C} \\ Y_{C} \\ 1 \end{bmatrix}}{< {\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.}}$
 32. The computer readable storage medium of claim 28, wherein the attribute values comprise at least one of color values and texture values.
 33. The computer readable storage medium of claim 28, further comprising instructions that cause one or more processors to store in a z-buffer a set of z-values associated with the pixels, and to compare a z-value, z_(c), of the current pixel with a corresponding z-value, z_(b), of a z-buffer to determine whether each pixel within the rectangular area is visible and selectively renders each pixel of the rectangular area that is visible and that falls within the triangular area.
 34. The computer readable storage medium of claim 28, further comprising instructions that cause one or more processors to issue a command that specifies vertices of the triangular area.
 35. The computer readable storage medium of claim 28, further comprising instructions that cause one or more processors to: buffer the vertices of the triangular area to be rendered; process the vertices to compute bounding data that define the dimensions of the rectangular area; and process the bounding data and evaluating coordinates associated with one or more of the pixel values of the rectangular area to selectively render pixels that fall within the triangular area.
 36. The computer readable storage medium of claim 35, further comprising instructions that cause one or more processors to: receive the buffered vertices and process the vertices to compute linear coefficients for a set of linear equations that describe edges of the triangular area, and process the vertices to compute linear coefficients for a set of linear equations that describe one or more attributes associated with the triangular area, wherein the bounding data and the coefficients are processed in accordance with the sets of linear equations to render the pixels that fail within the triangular area.
 37. The computer readable storage medium of claim 28, wherein the instructions are contained in a wireless communication device.
 38. The computer readable storage medium of claim 28, wherein the one or more processors are integrated circuits.
 39. The computer readable storage medium of claim 28, further comprising instructions that cause one or more processors to store to a cache memory at least a portion of the pixels within the rectangular area, wherein the cache memory has a block size, and to define the rectangular area as a function of the block size of the cache memory.
 40. The computer readable storage medium of claim 28, wherein sequentially evaluate coordinates associated with the pixels of each line of pixels to determine whether the pixels fall within the triangular area comprises evaluating the pixels in a rightward and downward fashion. 